Question: The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $x(x+5)$, where $x$ is a positive integer. If one of the integers is 50, what is the smallest possible value of the other one?
We know that $\gcd(m,n) \cdot \mathop{\text{lcm}}[m,n] = mn$ for all positive integers $m$ and $n$.  Hence, in this case, the other number is \[\frac{(x + 5) \cdot x(x + 5)}{50} = \frac{x(x + 5)^2}{50}.\]To minimize this number, we minimize $x$.

We are told that the greatest common divisor is $x + 5$, so $x + 5$ divides 50.  The divisors of 50 are 1, 2, 5, 10, 25, and 50.  Since $x$ is a positive integer, the smallest possible value of $x$ is 5.  When $x = 5$, the other number is $5 \cdot 10^2/50 = 10$.

Note that that the greatest common divisor of 10 and 50 is 10, and $x + 5 = 5 + 5 = 10$.  The least common multiple is 50, and $x(x + 5) = 5 \cdot (5 + 5) = 50$, so $x = 5$ is a possible value.  Therefore, the smallest possible value for the other number is $\boxed{10}$.